Marooned in Hilbert Space
Copyright © 2001, David A. Epstein.
All Rights Reserved.
David was about to be launched into a Hilbert space, the infinite dimensional home of the quantum world. After being finessed into this complete, inner product metric space, he would float in his space suit and use wave generation controllers to explore its glaring depth and beauty.
Various wave operators and gauge transformations sent him into progressively higher dimensional spaces. First, starting as a four-dimensional being, three spatial dimensions and one of time, he bifurcated into a sentient 8D being. Then, he bifurcated into a 16D being. Concurrent existence occurred with each bifurcation. For example, once the 4D David branched into the 8D David, both of these beings simultaneously existed in their respective worlds. This process was continuously repeated, tending towards the creation of an infinite number of Davids.
Based upon a process of “borrowed energy”, an anti-entropic breeding procedure created an increased level of energy for the bifurcated being. The 8D David borrowed energy from the 4D David, and thus had more energy than his predecessor; in turn, the 16D David, borrowing from the 8D David, had more energy. Because the dimension growth rate increased more rapidly than the corresponding increase of energy, the energy sequence converged, meaning the David existing in infinite dimensions had a finite amount of energy.
Each bifurcation was accompanied by a time contraction. The time to create a new dimensional being was progressively cut in half. The leap from four to eight dimensions took one second, from eight to sixteen a half of a second, and so forth. This geometric series took two seconds to propel him into infinite dimensions. At that point, time ceased to exist, but merely for a moment. As the force of contraction reached its peak, time recoiled into its complete length. This decompression of compact time helped to propel infinite David through the Hilbert space.
It was a cornucopia of geometric and topological delights. He saw the terrain stretch in every conceivable direction out to the horizon. There was an unending series of dimensions intersecting at right angles. As he moved in one direction, he felt himself move in another. It gave him a feeling of motion sickness, but he was too much in awe to let it distract him.
Vast planes of topographical tapestries were laid out for frontier footsteps. He floated over rolling hills draped in velvet and embroidered with emerald objects. The vibrations from these objects somehow penetrated his suit and reached his senses. He did not hear ordinary sounds, but deeply felt intricate, multi-textured musical patterns.
In the valleys were all types of quantum particles frolicking in bubbling orange meadows and blue open fields. While he longed to join them, he knew that he could not land on the boiling surface. He felt its intense heat. Running streams of pions reflected the subtle harmonies of quark symmetries underlying nuclear particles. He saw some charges trapped in the field lines of the nuclear particles, but kept a safe distance from them.
Against this backdrop, the sky conducted a symphony of multicolored vortices of intense energy. As this plasma-like substance spiraled into the centers of these vortices, new energy appeared to emerge from them. This kaleidoscopic display of apparent energy recycling was a sheer wonder of quantum ecology.
“Can it get any better than this?” thought David. In a respect, he was like a pantheistic god. He felt his existence behind him, below him, within him.
While the wave-generated Davids meandered in their respective dimensional spaces, infinite-dimensional David continued to travel through the Hilbert space. He felt the force of a Cauchy sequence converging upon him, keeping him confined within his current space. This was an algebraic sequence of numbers whose elements clustered together no matter how far out in the sequence they were located. David was being gently massaged by one of these clusters. This immediate reality of containment, of being there and only there, is what made him feel complete.
Orthogonal vectors, linear directional pathways adjoined at right angles, were projected in this space. David felt these vectors as he slowly moved towards them. One of the vectors flew right past him, like Poseidon throwing a lightning bolt. He looked intently in that direction as he realized just how close it came to nailing him.
Offsetting forces occurred in the Hilbert Space. Various linear operators methodically acted upon it, transforming it into a festival of quantum events. On the other hand, David experienced the vibrations of annihilator operators destroying their targeted quantum physical states. This left him in a slight panic; he began vibrating in different directions.
It took a few seconds for him to setting down. Once he did, he floated slowly through the space. He was intrigued by all of the displays of quantum events. David viewed all the physical observables such as position, momentum, energy and the spin of an orbiting electron.
Each of these observables had a corresponding self-adjoint operator acting upon the Hilbert Space. In turn, each operator corresponded to a spectrum of discrete values. Only elements of the spectrum could be seen. These elements danced with cherubic delight, with flaming colors and sparkling sounds, while non-elements, condemned to eternal phantom existence, looked on with great envy.
In a series of dynamical steps, a Hermitian operator approached him. He tried running away, but resistance proved futile. This self-adjoint energy operator treated him like he was a vector in a state of bondage. For the time being, David was in quantum boot camp being drilled into submission; however, he would not be treated this way for long, only long enough to condition him to the inner workings of operator mechanics.
During the next exhibit, he experienced the soothing symmetries of Lie Algebras. Then, he savored the delights of compact, bounded Hilbert-Schmidt operators. They squeezed and tossed him against their cushioned walls. He thought about some of the rides he rode at an amusement park when he was a kid. At the same time, he was astounded how these physical principles could evoke such rich spectral recollections of dormant memories.
“Oooooh. What happened?” he cried out as he went flying off. He was traveling at a very fast speed, unaware that he had been hit by a photon. A probing beam of light had been emitted to visibly mark his position.
David saw another beam of light approach him. It now dawned on him what had happened. He braced himself for the collision, but no matter how hard he tried, he felt the photonic sting, even though his protective suit absorbed much of the impact. The light sought to pinpoint his location, but it had Heisenberg written all over it and rapidly propelled him out of the Hilbert Space.
He was scared about what would happen, yet he felt exhilarated traveling so rapidly through Microspace. The subatomic particles whizzed right by him as if they were blind dates who had no intention of going out with him again. It was a continuous blur of meshing colors, abstract patterns and visual harmonics. He had a flashback to the Museum of Modern Art in New York City where he viewed the works of Jackson Pollack and other contemporary artists. It gave him a very warm feeling, but like all memories evoked from a fast-moving medium, this flashback got caught in a time dilation and just stopped ticking.